行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}{-}1 & {-}1 & {-}2 & 2 \\ 2 & 2 & {-}1 & 2 \\ 2 & {-}2 & {-}1 & 2 \\ 2 & 1 & 2 & {-}2\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 & 2 \\ 2 & 2 & {-}1 & 2 \\ 2 & {-}2 & {-}1 & 2 \\ 2 & 1 & 2 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [2, 2, 2, -2]

\( \qquad\quad = (-1)^{4-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}1 \\ 2 & {-}2 & {-}1 \\ 2 & 1 & 2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 \\ 2 & {-}2 & {-}1 \\ 2 & 1 & 2\end{array}\,\right| + (-1)^{4-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 \\ 2 & 2 & {-}1 \\ 2 & 1 & 2\end{array}\,\right| + (-1)^{4-4}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 \\ 2 & 2 & {-}1 \\ 2 & {-}2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times2\times(-24) + (-1)^{4-2}\times2\times(-3) + (-1)^{4-3}\times2\times5 + (-1)^{4-4}\times(-2)\times20 \)

\( \qquad\quad = -8 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}1 \\ 2 & {-}2 & {-}1 \\ 2 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, -1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}2 \\ 2 & 1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times6 + (-1)^{3-2}\times(-1)\times(-2) + (-1)^{3-3}\times2\times(-8) \)

\( \qquad\quad = -24 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}2 \\ 2 & 1\end{array}\,\right| = 2\times1 - (-2)\times2 = 6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & 1\end{array}\,\right| = 2\times1 - 2\times2 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & {-}2\end{array}\,\right| = 2\times(-2) - 2\times2 = -8 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 \\ 2 & {-}2 & {-}1 \\ 2 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, -1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}2 \\ 2 & 1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times6 + (-1)^{3-2}\times(-1)\times1 + (-1)^{3-3}\times2\times4 \)

\( \qquad\quad = -3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}2 \\ 2 & 1\end{array}\,\right| = 2\times1 - (-2)\times2 = 6 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times2 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}2\end{array}\,\right| = (-1)\times(-2) - (-1)\times2 = 4 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 \\ 2 & 2 & {-}1 \\ 2 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, -1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & 1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times(-2) + (-1)^{3-2}\times(-1)\times1 + (-1)^{3-3}\times2\times0 \)

\( \qquad\quad = 5 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & 1\end{array}\,\right| = 2\times1 - 2\times2 = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 1\end{array}\,\right| = (-1)\times1 - (-1)\times2 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 2\end{array}\,\right| = (-1)\times2 - (-1)\times2 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & {-}1 & {-}2 \\ 2 & 2 & {-}1 \\ 2 & {-}2 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, -1, -1]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & {-}2\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}2\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times(-8) + (-1)^{3-2}\times(-1)\times4 + (-1)^{3-3}\times(-1)\times0 \)

\( \qquad\quad = 20 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 2 & {-}2\end{array}\,\right| = 2\times(-2) - 2\times2 = -8 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & {-}2\end{array}\,\right| = (-1)\times(-2) - (-1)\times2 = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & {-}1 \\ 2 & 2\end{array}\,\right| = (-1)\times2 - (-1)\times2 = 0 \)